Three contenders - Mark Harper, Andrea Leadsom and Esther McVey - have been knocked out, in the secret ballot held in the House of Commons.
Johnson has told fellow Brexiteers that he will leave open the radical possibility of suspending Parliament to force through a "no-deal" Brexit if negotiations with Brussels fail, rather than again delay Brexit, The Times of London reported Thursday.
Johnson meanwhile has picked up the support of Trump, who said both he and Hunt would be good in the top job.
"I am delighted to win the first ballot, but we have a long way to go", Johnson wrote on Twitter after his victory in the first round.
Johnson, a former foreign secretary and leading Brexit campaigner, secured 114 of the 313 votes cast in the round, which reduced the field of candidates from 10 to seven.
The victor of the leadership race is set to become the UK's next Prime Minister after Theresa May announced her resignation on the 24th May.
Subsequent ballots are scheduled to take place on 18, 19 and 20 June to whittle down the contenders one by one until only two are left.
Mr Stewart, who has criticised Mr Johnson for making promises on Brexit he can not keep, voiced optimism about his campaign: "This is awesome - we're getting some real momentum here ... it's increasingly clear it's me against Boris".
After weeks of hustings around the country, the 160,000 grass-roots Conservative party members pick the victor, with the result announced in the week beginning July 22.
The former Conservative Prime Minister said anyone even considering this was guilty of "hypocrisy on a gold-plated stand". It's increasingly clear it's me against Boris.
"After three years and two missed deadlines, we must leave the European Union on October 31", Johnson, 54, said then.
All the Conservative contenders say if they emerge as the victor, they will renegotiate the contentious Brexit withdrawal agreement Theresa May struck with Brussels last November after two years of ill-tempered haggling. As with his party rivals, Johnson hasn't provided any details on how he would overcome the parliamentary majority determined to block Britain leaving without a deal, and how he plans to coax European Union leaders to reopen talks. No deals have yet been done, the sources said.